On 13 Feb 2014, at 22:23, LizR wrote:
On 14 February 2014 07:49, Bruno Marchal <[email protected]> wrote:
Liz, and others,
On 13 Feb 2014, at 10:04, LizR wrote:
On 13 February 2014 21:38, Bruno Marchal <[email protected]> wrote:
If I reported that there was a flying pig, wouldn't comp just
explain, "That's the way arithmetic looks from inside."?
Why? No. Not at all.
You must (using G & Co.) looks at the way arithmetic looks from
inside, and if you find the flying pig, then yes, you can say that
comp explains the flying pig, but if you see a white rabbit instead
(in the arithmetic), you can say that comp does not explain the
flying pig, and might be false in case you don't find the white
rabbit in nature.
This sounds like my sort of science!
It is the scientist sort of science, yes.
"One pill makes you larger... And one pill makes you small..."
Unfortunately, in the *realist* fairy tales, sometimes you don't
have the magic wand, nor magic pills, and you have to empty an ocean
with a tea spoon, if not a sieve, and be confronted with ten
thousand Monsters if not grin without cat!
We say that "reality" kick back, and it it is an euphemism. And that
explains probably why science asks for some amount of works. Even
tedious one!
Well, just to prepare you for 8, well, no, 7 exercises.
And more official definitions.
Definition: A Kripke multiverse (W, R) is a non empty set W, with a
binary relation R.
That's all you need to know, about a Kripke multiverse.
Conventionally we use the greek letter for its element, alpha,
beta, ... and we call them world.
OK?
Exercise: give examples of the simplest Kripke multiverses possible.
W = { alpha } R = alphaRalpha (R is true or false for accessibility
of alpha from itself - although I'm not sure how a universe can't be
accessible from itself, to be honest).
Keep in mind that we are doing math, and a "universe", or "world" is
any element of an non empty set, and "accessible" is any binary
relation.
Then we will see that the logical modal theory (axioms and rules, and
theorems) can be characterized by Kripke multiverse and their
relation of accessibility. That is useful to find conterexamples, to
see that some modal formula are independent of others.
You can also keep in mind, that we are not interested in modal logic
per se. We will model (and justify somehow) a notion of rational
believability by Gödel beweisbar *arithmetical" predicate. That
predicate is Turing complete, and is a name of the machine, it is a
universal numbers, as I define them. It is the code of an arithmetical
interpreter: you give him i and j and it computes phi_i(j).
Do you still remember what the phi_i are? Just in case you did, phi_i
is the ith partial computable function corresponding to the ith code
in a universal programming languages, or systems, or numbers.
It just happens that modal logic shortened a lot the interview of the
universal Gödel-Löbian machine, especially on the intensional nuances.
Solution:
W cannot be empty (by the definition above), so the simplest one is
probably the singleton {alpha}, with the empty binary relation---
alpha does not access to any world, not even itself.
Perhaps the next simplest one is {alpha}, with alpha R alpha. Alpha
access to itself.
OK?
Oh, OK, I assumed alpa R alpha was "equivalently simple" whether R
was true of false. (Why is inaccessible considered simpler? In a way
I'd consider self-accessible simpler!)
Oh! It is simpler as the empty set is simpler than a singleton, or
that 0 is simpler than 1 (which of course is not quite true).
The empty relation is given by an empty set { }, even on a non empty
multiworld. Your relation is {(alpha, alpha)}, which needs more K than
"{ }".
But Kripke multiverse can be illuminated. By this I mean that we can
associate a truth value (t, f, or 1, 0 ...) to all propositional
letters, and this for each world. In english: we just illuminate the
multiverse by telling the truth value of the atomic propositions (p,
q, r, ...) in each world. (keep in mind that the goal is to find
counterexample in modal reasoning).
OK
Good.
We stipulate also that all worlds obeys classical propositional
logic (CPL).
In particular, if A is true at alpha, and if B is true at alpha,
then (A & B) is true at alpha. For example. You cannot have a world
with A false, but (A & B) true. Etc. OK?
OK
OK.
Let us consider only one propositional letters. As a matter of fact,
although the possible truth value of []p does not depend on p, it is
independent of the value of q, so, to start, we can play with only
one propositional letter, as most formula have only one
propositional letters occurring in them.
But how to decide the truth of a modal formula, with occurrence of
[]p and <>p?
All you have to keep in mind is that (Kripke semantics):
[]p is true in alpha <=> p is true in all the worlds accessible
from alpha
(dually, you have already seen this)
<>p is true in alpha <==> there is at least one world accessible
from alpha where p is true.
So is []p true in the simplest multiverse with nowhere accessible?
(It looks like both []p and ~[]p are true!)
No.
It looks like []p and []~p is true.
Be very careful with this.
[]~p is true in alpha means that ~p is true in all worlds accessible
from alpha. It is "necessary" *relatively* to alpha.
~[]p is true in alpha means that there is some world beta where p is
false.
The confusion between atheism and agnosticism is a confusion between
[]~ (atheists, accepting some definition), with agnostic (~[]).
OK?
<>p is false because there isn't an accessible world.
Yes. Cul-de-sac. No hope.
OK?
It is really the same semantics as Leibniz, but relativized on each
relatively accessible worlds.
This determines the truth value of []p and <>p in the illuminated
multiverse. To see if []p is true in a world, just look at those
world accessed by alpha, and see if p is true at them. If alpha
access ten worlds, you have to look at those ten worlds. If alpha
access 0 worlds, you have zero verification to do, making the truth
of []p vacuously true (or use []p = ~<>~p).
OK that answers the question. Of course []p = ~<>~p so <>~p says
there is a world accessible from alpha where p is false. With no
worlds accessible that is false, so ~<>~p is true.
and so []p is true, no matter what p is, or what his illumination t,
or f is.
OK?
OK
Now, ask any question if anything remains unclear up to here, before
trying the 7) exercises.
But here is the exercise (we work with only one propositional letter)
First illuminate the two simplest multiverses above, that is {alpha}
with no accessibility relation, and {alpha} with alpha R alpha. That
should not be long, given that we restrict ourself to only one
propositional variable p, and have only one world.
Well, we get { p=t } and { p=f } regardless of the accessibility
relations. (If that's how you write it)
Well ... OK.
More precisely we get
1) {alpha}, with R = { } with p=t in alpha
2) {alpha}, with R = { } with p=f in alpha
and
3) {alpha}, with R = {(alpha R alpha) } with p=t in alpha
4) {alpha}, with R = {(alpha R alpha) } with p=f in alpha
Which of those propositions are true of false in alpha, in the
illuminated simplest multiverses.
And which one are law (meaning true in all worlds, but true for all
valuation of p, that is valid with A = p, but also with A = ~p)
1) []A -> A
This is true in alpha R alpha (because it's just a Leibniz type world)
Very good. It is a law there.
. []p is "vacuously true" in "alpha" (the disconnected multiverse) -
as you said above - so []p -> p is false, because []p is true
regardless of p.
OK. It is not a law, but it might still be true in some circumstances.
Give me which among 1), 2), 3), 4), above.
2) []A -> [][]A
This is true in alpha R alpha, and in alpha I guess it's true too,
because vacuously true implies vacuously true?
Exact.
3) <>A -> []<>A
true in alpha R alpha again, because there's only one world to
consider so <>A is equivalent to []A in this case (isn't it?)
Well seen!
not true in alpha because []<>A is vacuously true regardless of <>A
- I think
Not correct. You jump to hastily.
in your language the answer is:
true in alpha, because []<>A is vacuously true, so that <>A -> []<>A
is vacuously true too (as "p -> q" is false only if q is false and p
is true). The type of <>A -> []<>A is really f -> t, which is as much
tautological than f -> A, and A -> f, for any A.
4) []A -> <>A
Well I think this is true for reason given above.
You begin to try to go to quickly. I have some doubt that []A -> <>A
can be a law in a cul-de-sac world, like poor alpha, with R = { }.
5)A -> []<>A
True in alpha R alpha.
OK.
In alpha not true because []<>A is always true and A isn't
Not a law. OK. Again there are case where it is true, like when A is
true.
6) <>A -> ~[]<>A
False in alpha R alpha, surely? With one world, <>A -> []<>A (above)
Correct.
Not true in alpha because ~{}<>A is vacuously false regardless of <>A
Unfortunately as much as ~{}<>A is vacuously false regardless of <>A,
as you say, we are interested in
<>A -> ~[]<>A and in poor alpha (case 1) 2)) <>A is *also vacuously
false, so that we are in the f -> f, case, which is, vacuously or not,
always true.
7) []([]A -> A) -> []A
True in alpha R alpha I think.
A law? True in both 3), 4) ?
And vacuously true in alpha because both sides of the rightmost ->
have to be true.
Correct. It is law. True in 1) 2).
Not sure if that means it's implied though...
8) []([](A -> []A) -> A) -> A
Not true in alpha because to the left of rightmost -> is vacuously
true regarldess of A.
OK. Not a law.
Don't know about alpha R alpha because my head exploded...
Take a break. I said "one a at a time" !
Let me solve one case, to illustrate. Let us look at []A -> <>A, in
the illuminated multiverse {alpha}, with empty R, and with p true in
alpha.
Well, what about []A?
Alpha is a cul-de-sac world, so we have seen that []A must be true
(if not <>~A has to be true), so []A is true and in particular []p
is true (whatever the value of p is). What about <>A ? well this say
that there is some beta accessible from alpha in which A is true.
But alpha is a cul-de-sac world, so there is no such world, and so
<>A is false, whatever A is (notably p or ~p). So []A is always
true, and <>A is always false, so []A -> <>A is always false (by
CPL), whatever illumination is chosen (p or ~p true at alpha).
OK?
OK
It is just CPL used in the world alpha, with the value of "[]A"
determined by the Kripke semantics.
Can you see the truth value of the 7 other formula in those simplest
multiverses. I think it is a good training, and a good way to
demystify the difficulties. Tell me what. You can do one formula at
a time, in 8 posts.
I did them all (or more likely I didn't) before I got to this
point..........see above..........
Yeah, I should have said "one at a time" earlier :)
Very good work.
I give you more time for Grzegorczyk []([](A -> []A) -> A) -> A
And just one supplementary exercise. What can be said about
[](A -> B) -> ([]A -> []B), in those simple multiworlds structure?
This formula has two (meta) variables A and B.
Is that a law?
Bruno
http://iridia.ulb.ac.be/~marchal/
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